Only went to the same class is natural. My friend and I ...
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MustAskIt would be too awkwardI don't think it would. There's nothing wrong with correcting your own speech in conversation, especially if you're still learning the language.
MustAskSo "me and my friend" sounds somehow odd and incorrect?It's not odd; in fact, it's quite common among careless native speakers. Nonetheless, it's ungrammatical, and I don't suggest making a habit of it.
MustAsk " Me and my friend... we uhmm , we went to the same class..."I would be wincing and saying to myself "My friend and I! My friend and I!"